Comptes Rendus
no. 3
Numerical Analysis
A superconvergent projection method for nonlinear compact operator equations
[Une méthode de projection superconvergente pour les équations d'opérateurs nonlinéaires compacts]

Présenté par : Philippe G. Ciarlet

Laurence Grammont1 ; Rekha Kulkarni2
1 Laboratoire de mathématiques de l'université de Saint-Étienne, 23, rue du Dr. Paul Michelon, 42023 Saint Étienne cedex 2, France
2 Department of Mathematics, Indian Institute of Technology, Powai, Mumbai 400076, India
Comptes Rendus. Mathématique, Volume 342 (2006) no. 3, pp. 215-218.

Nous proposons une méthode, basée sur une projection, pour approcher les points fixes localement uniques d'un opérateur compact. Cette méthode présente un avantage par rapport aux méthodes de Galerkin et de Galerkin itérée étudiées par K.E. Atkinson and F.A. Potra : on n'a pas besoin de conditions supplémentaires pour obtenir la superconvergence de la solution approchée vers la solution exacte.

We propose a method based on projections for approximating fixed points of a compact nonlinear operator. Under the same assumptions as in the Galerkin method, the proposed solution is shown to converge faster than the Galerkin solution.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2005.11.011

Laurence Grammont 1 ; Rekha Kulkarni 2

1 Laboratoire de mathématiques de l'université de Saint-Étienne, 23, rue du Dr. Paul Michelon, 42023 Saint Étienne cedex 2, France
2 Department of Mathematics, Indian Institute of Technology, Powai, Mumbai 400076, India 
@article{CRMATH_2006__342_3_215_0,
     author = {Laurence Grammont and Rekha Kulkarni},
     title = {A superconvergent projection method for nonlinear compact operator equations},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {215--218},
     publisher = {Elsevier},
     volume = {342},
     number = {3},
     year = {2006},
     doi = {10.1016/j.crma.2005.11.011},
     language = {en},
}
TY  - JOUR
AU  - Laurence Grammont
AU  - Rekha Kulkarni
TI  - A superconvergent projection method for nonlinear compact operator equations
JO  - Comptes Rendus. Mathématique
PY  - 2006
SP  - 215
EP  - 218
VL  - 342
IS  - 3
PB  - Elsevier
DO  - 10.1016/j.crma.2005.11.011
LA  - en
ID  - CRMATH_2006__342_3_215_0
ER  - 
%0 Journal Article
%A Laurence Grammont
%A Rekha Kulkarni
%T A superconvergent projection method for nonlinear compact operator equations
%J Comptes Rendus. Mathématique
%D 2006
%P 215-218
%V 342
%N 3
%I Elsevier
%R 10.1016/j.crma.2005.11.011
%G en
%F CRMATH_2006__342_3_215_0
Laurence Grammont; Rekha Kulkarni. A superconvergent projection method for nonlinear compact operator equations. Comptes Rendus. Mathématique, Volume 342 (2006) no. 3, pp. 215-218. doi : 10.1016/j.crma.2005.11.011. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2005.11.011/

[1] K.E. Atkinson; I. Graham; I. Sloan Piecewise continuous collocation for integral equations, SIAM J. Numer. Anal., Volume 20 (1983), pp. 172-186

[2] K.E. Atkinson; F.A. Potra Projection and iterated projection methods for nonlinear integral equations, SIAM J. Numer. Anal., Volume 24 (1987) no. 6, pp. 1352-1373

[3] M.A. Krasnosel'skii; G.M. Vainikko; P.P. Zabreiko; Y. Rutitskii; V. Stetsenko Approximate Solution of Operator Equation, P. Noordhoff, Groningen, 1972

[4] M.A. Krasnosel'skii; P.P. Zabreiko Geometrical Methods of Nonlinear Analysis, Springer-Verlag, Berlin, 1984

[5] R.P. Kulkarni A superconvergence result for solutions of compact operator equations, Bull. Austral. Math. Soc., Volume 68 (2003), pp. 517-528

[6] R.P. Kulkarni A new superconvergent projection method for approximate solutions of eigenvalue problems, Numer. Funct. Anal. Optim., Volume 24 (2003), pp. 75-84

[7] G.M. Vainikko Galerkin's perturbation method and general theory of approximate methods for non-linear equations, USSR Comput. Math. Math. Phys., Volume 7 (1967) no. 4

  • Payel Das; Kapil Kant; B. V. Ratish Kumar Modified Galerkin method for Volterra-Fredholm-Hammerstein integral equations, Computational and Applied Mathematics, Volume 41 (2022) no. 6 | DOI:10.1007/s40314-022-01945-9
  • Rekha P. Kulkarni; Gobinda Rakshit DISCRETE MODIFIED PROJECTION METHODS FOR URYSOHN INTEGRAL EQUATIONS WITH GREEN’S FUNCTION TYPE KERNELS, Mathematical Modelling and Analysis, Volume 25 (2020) no. 3, p. 421 | DOI:10.3846/mma.2020.11093
  • C. Allouch; D. Sbibih; M. Tahrichi Superconvergent product integration method for Hammerstein integral equations, Journal of Integral Equations and Applications, Volume 31 (2019) no. 1 | DOI:10.1216/jie-2019-31-1-1
  • Moumita Mandal; Gnaneshwar Nelakanti Superconvergence Results for Volterra-Urysohn Integral Equations of Second Kind, Mathematics and Computing, Volume 655 (2017), p. 358 | DOI:10.1007/978-981-10-4642-1_31
  • Laurence Grammont; Rekha P. Kulkarni; T.J. Nidhin Modified projection method for Urysohn integral equations with non-smooth kernels, Journal of Computational and Applied Mathematics, Volume 294 (2016), p. 309 | DOI:10.1016/j.cam.2015.08.020
  • Rekha P. Kulkarni; T.J. Nidhin Approximate solution of Urysohn integral equations with non-smooth kernels, Journal of Integral Equations and Applications, Volume 28 (2016) no. 2 | DOI:10.1216/jie-2016-28-2-221
  • C. Allouch; D. Sbibih; M. Tahrichi Superconvergent Nyström and degenerate kernel methods for Hammerstein integral equations, Journal of Computational and Applied Mathematics, Volume 258 (2014), p. 30 | DOI:10.1016/j.cam.2013.08.025
  • Laurence Grammont; Rekha P. Kulkarni; Paulo B. Vasconcelos Modified projection and the iterated modified projection methods for nonlinear integral equations, Journal of Integral Equations and Applications, Volume 25 (2013) no. 4 | DOI:10.1216/jie-2013-25-4-481

Cité par 8 documents. Sources : Crossref

Commentaires - Politique